Invariants
| Level: | $24$ | $\SL_2$-level: | $24$ | Newform level: | $576$ | ||
| Index: | $288$ | $\PSL_2$-index: | $144$ | ||||
| Genus: | $9 = 1 + \frac{ 144 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 8 }{2}$ | ||||||
| Cusps: | $8$ (of which $2$ are rational) | Cusp widths | $12^{4}\cdot24^{4}$ | Cusp orbits | $1^{2}\cdot2^{3}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| Analytic rank: | $2$ | ||||||
| $\Q$-gonality: | $4 \le \gamma \le 6$ | ||||||
| $\overline{\Q}$-gonality: | $4 \le \gamma \le 6$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 24I9 |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 24.288.9.4097 |
Level structure
| $\GL_2(\Z/24\Z)$-generators: | $\begin{bmatrix}5&14\\8&13\end{bmatrix}$, $\begin{bmatrix}9&10\\16&9\end{bmatrix}$, $\begin{bmatrix}11&6\\12&1\end{bmatrix}$, $\begin{bmatrix}19&22\\20&13\end{bmatrix}$, $\begin{bmatrix}21&16\\8&21\end{bmatrix}$ |
| $\GL_2(\Z/24\Z)$-subgroup: | $C_4.D_4^2$ |
| Contains $-I$: | no $\quad$ (see 24.144.9.mo.1 for the level structure with $-I$) |
| Cyclic 24-isogeny field degree: | $4$ |
| Cyclic 24-torsion field degree: | $32$ |
| Full 24-torsion field degree: | $256$ |
Jacobian
| Conductor: | $2^{40}\cdot3^{18}$ |
| Simple: | no |
| Squarefree: | no |
| Decomposition: | $1^{9}$ |
| Newforms: | 36.2.a.a$^{3}$, 144.2.a.a, 576.2.a.b, 576.2.a.c, 576.2.a.d, 576.2.a.g, 576.2.a.h |
Models
Canonical model in $\mathbb{P}^{ 8 }$ defined by 21 equations
| $ 0 $ | $=$ | $ x v + y t $ |
| $=$ | $t u - u^{2} - r s$ | |
| $=$ | $x r - z v - w u$ | |
| $=$ | $x s - z t + z u + w v$ | |
| $=$ | $\cdots$ |
Singular plane model Singular plane model
| $ 0 $ | $=$ | $ x^{12} + x^{8} y^{4} + 18 x^{8} y^{2} z^{2} - x^{4} y^{8} - 36 x^{4} y^{6} z^{2} + 108 x^{4} y^{4} z^{4} + \cdots + 216 y^{6} z^{6} $ |
Rational points
This modular curve has 2 rational cusps but no known non-cuspidal rational points. The following are the coordinates of the rational cusps on this modular curve.
| Canonical model |
|---|
| $(0:0:0:0:0:1:1:-1:1)$, $(0:0:0:0:0:-1:-1:-1:1)$ |
Maps to other modular curves
Map of degree 2 from the canonical model of this modular curve to the canonical model of the modular curve 24.72.4.bl.1 :
| $\displaystyle X$ | $=$ | $\displaystyle -y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle -x$ |
| $\displaystyle Z$ | $=$ | $\displaystyle -r$ |
| $\displaystyle W$ | $=$ | $\displaystyle -s$ |
Equation of the image curve:
| $0$ | $=$ | $ 48X^{2}-6Y^{2}+Z^{2}-W^{2} $ |
| $=$ | $ 18XY^{2}+XZ^{2}+2YZW-XW^{2} $ |
Map of degree 1 from the canonical model of this modular curve to the plane model of the modular curve 24.144.9.mo.1 :
| $\displaystyle X$ | $=$ | $\displaystyle y$ |
| $\displaystyle Y$ | $=$ | $\displaystyle \frac{1}{2}z$ |
| $\displaystyle Z$ | $=$ | $\displaystyle \frac{1}{12}t$ |
Equation of the image curve:
| $0$ | $=$ | $ X^{12}+X^{8}Y^{4}+18X^{8}Y^{2}Z^{2}-X^{4}Y^{8}-36X^{4}Y^{6}Z^{2}+108X^{4}Y^{4}Z^{4}-Y^{12}+18Y^{10}Z^{2}-108Y^{8}Z^{4}+216Y^{6}Z^{6} $ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.144.4-24.bl.1.17 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{5}$ |
| 24.144.4-24.bl.1.25 | $24$ | $2$ | $2$ | $4$ | $1$ | $1^{5}$ |
| 24.144.4-24.ch.1.28 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{5}$ |
| 24.144.4-24.ch.1.38 | $24$ | $2$ | $2$ | $4$ | $0$ | $1^{5}$ |
| 24.144.5-24.o.1.3 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
| 24.144.5-24.o.1.17 | $24$ | $2$ | $2$ | $5$ | $1$ | $1^{4}$ |
This modular curve is minimally covered by the modular curves in the database listed below.
| Covering curve | Level | Index | Degree | Genus | Rank | Kernel decomposition |
|---|---|---|---|---|---|---|
| 24.576.17-24.fq.1.21 | $24$ | $2$ | $2$ | $17$ | $5$ | $1^{8}$ |
| 24.576.17-24.gq.1.11 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 24.576.17-24.wq.1.21 | $24$ | $2$ | $2$ | $17$ | $4$ | $1^{8}$ |
| 24.576.17-24.yd.1.10 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 24.576.17-24.bmq.1.9 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 24.576.17-24.bms.1.15 | $24$ | $2$ | $2$ | $17$ | $4$ | $1^{8}$ |
| 24.576.17-24.bng.1.9 | $24$ | $2$ | $2$ | $17$ | $3$ | $1^{8}$ |
| 24.576.17-24.bni.1.11 | $24$ | $2$ | $2$ | $17$ | $4$ | $1^{8}$ |
| 24.576.17-24.btg.1.6 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.576.17-24.btg.2.8 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.576.17-24.bth.1.5 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.576.17-24.bth.2.6 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.576.17-24.bti.1.8 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.576.17-24.bti.2.5 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.576.17-24.btj.1.6 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.576.17-24.btj.2.7 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.576.17-24.btk.1.6 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.576.17-24.btk.2.6 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.576.17-24.btl.1.8 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.576.17-24.btl.2.3 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.576.17-24.btm.1.4 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.576.17-24.btm.2.8 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.576.17-24.btn.1.3 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 24.576.17-24.btn.2.6 | $24$ | $2$ | $2$ | $17$ | $2$ | $2^{4}$ |
| 48.576.19-48.jh.1.14 | $48$ | $2$ | $2$ | $19$ | $2$ | $2^{3}\cdot4$ |
| 48.576.19-48.jh.2.14 | $48$ | $2$ | $2$ | $19$ | $2$ | $2^{3}\cdot4$ |
| 48.576.19-48.nr.1.6 | $48$ | $2$ | $2$ | $19$ | $2$ | $2^{3}\cdot4$ |
| 48.576.19-48.nr.2.6 | $48$ | $2$ | $2$ | $19$ | $2$ | $2^{3}\cdot4$ |
| 48.576.19-48.po.1.6 | $48$ | $2$ | $2$ | $19$ | $5$ | $1^{10}$ |
| 48.576.19-48.po.2.2 | $48$ | $2$ | $2$ | $19$ | $5$ | $1^{10}$ |
| 48.576.19-48.pr.1.10 | $48$ | $2$ | $2$ | $19$ | $5$ | $1^{10}$ |
| 48.576.19-48.pr.2.2 | $48$ | $2$ | $2$ | $19$ | $5$ | $1^{10}$ |
| 48.576.19-48.qc.1.10 | $48$ | $2$ | $2$ | $19$ | $3$ | $1^{10}$ |
| 48.576.19-48.qc.2.2 | $48$ | $2$ | $2$ | $19$ | $3$ | $1^{10}$ |
| 48.576.19-48.qg.1.6 | $48$ | $2$ | $2$ | $19$ | $4$ | $1^{10}$ |
| 48.576.19-48.qg.2.2 | $48$ | $2$ | $2$ | $19$ | $4$ | $1^{10}$ |
| 48.576.19-48.ra.1.12 | $48$ | $2$ | $2$ | $19$ | $2$ | $2^{3}\cdot4$ |
| 48.576.19-48.ra.2.12 | $48$ | $2$ | $2$ | $19$ | $2$ | $2^{3}\cdot4$ |
| 48.576.19-48.ro.1.4 | $48$ | $2$ | $2$ | $19$ | $2$ | $2^{3}\cdot4$ |
| 48.576.19-48.ro.2.4 | $48$ | $2$ | $2$ | $19$ | $2$ | $2^{3}\cdot4$ |